Powers and Roots SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Think/Pair/Share, Quickwrite, Group Presentation, Visualize, Create Representations Dominique Wilkins Middle School is holding its annual school carnival. Each year, classes and clubs build game booths in the school gym. This year, the student council has asked Jonelle s math class for help in deciding what size the booths should be and how they should be arranged on the gym floor. The class will begin this work by reviewing some ideas about area. The rectangle above has been divided into squares. Assume that the length of each side of a small square is 1 cm. 1. Find the area of the rectangle. Explain your method. ACTIVITY 1.4 Before deciding on how to arrange the booths, the student council needs to know the area of the gym floor, so several class members went to the gym to measure the floor. They found that the length of the floor is 84 feet and the width of the floor is 50 feet. 2. Find the area of the gym floor and write an explanation to the student council telling what your answer means. Include units in your answer. The class is now going to focus on the area of squares since this is the shape of the base of many of the game booths. 3. Do you need to know both the length and width of a square to be able to determine its area? Explain your answer. Then draw a diagram as a model. Unit 1 Integers and Rational Numbers 29
ACTIVITY 1.4 Powers and Roots SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Create Representations, Look for a Pattern, Interactive Word Wall 4. Write a rule, in words, for finding the area of a square. This drawing shows the floor space of one of the carnival booths. It is a square with the length of one side labeled with the letter s. The s can be given any number value since the booths are going to be different sizes. S WRITING MATH Sometimes a dot is used as a symbol for multiplication. 3 2 = 6 and 3 2 = 6 are both ways to show 3 times 2 equals 6. 5. Complete this table of the areas of some different size booths. The length of a side in feet is represented by s, as in the drawing above. Include units for area in the last column. Length of Side (in feet) s = 3 s = 6 s = 8 s = 9 s = 11 Calculation Area of the Square ACADEMIC VOCABULARY A number written with an exponent is in exponential form. For each calculation in Question 5 you found the product of a number times itself. The product of a number times itself can be written with a base and an exponent. For example 5 5, can be written as 5 2. 6. Label the diagram using the terms base and exponent. 5 2 Numbers in this form are said to be in exponential form. 7. What are some ways in which this can be read? 30 SpringBoard Mathematics with Meaning TM Level 2
Powers and Roots ACTIVITY 1.4 SUGGESTED LEARNING STRATEGIES: Quickwrite, Self Revision/Peer Revision, Group Presentation, Interactive Word Wall, Look for a Pattern, Think/Pair/Share When you find the value of the expression 5 2, you are squaring the number 5. The number 25 is called the square of 5. 8. Why do you think the number 25 is called the square of 5? Draw a model in the space as part of your explanation. 9. The table below gives some booth sizes in exponential form. Write each expression as a product with the base used twice as a factor and in standard form. Exponential Form Product Using the Base as a Factor Twice 5 2 5 5 25 2 2 1 2 7 2 15 2 Standard Form 10. If you know the area of the floor of a square booth, how can you find the side length of the booth? Finding the side length of a square is called finding the square root. The square root of 36 is written as 36. 11. Complete each table. All numbers are in standard form. Number 2 3 4 5 7 10 n WRITING MATH The symbol is called a radical sign. It is used in expressions to show square roots. Square Number 4 9 16 25 49 100 n 2 Square Root Unit 1 Integers and Rational Numbers 31
ACTIVITY 1.4 Powers and Roots SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Quickwrite, Self Revision/Peer Revision, Group Presentation, Think/Pair/Share 12. What patterns do you notice in the tables you made in Question 11? 13. Think about what you have discovered about the area of a square and finding the side length of a square. Write a sentence to explain what the square root of a number means. 14. The carnival booth sizes are assigned according to the number of members in the club or class. To help decide what size booths are needed, complete this table. Area of Square Booth s Floor Side Length of Booth s Floor Club or Class Size 1 30 members 36 ft 2 31 60 members 8 ft 61 90 members 9 ft 91 120 members 121 ft 2 121 150 members 144 ft 2 The area of each booth floor you worked with in Question 14 is called a perfect square. 15. What do you think is meant by the term perfect square? Several of the classes would like to make booths with floors that have areas that are not perfect squares. One class wants to have a booth with a square floor that has an area of 18 ft 2. 16. Why would it be difficult to find the exact square root of 18? 32 SpringBoard Mathematics with Meaning TM Level 2
Powers and Roots ACTIVITY 1.4 SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Self Revision/Peer Revision, Think/Pair/Share, Create Representations, Identify a Subtask 17. Use this number line and the method described in Parts a f to determine the approximate side length of a square booth with a floor area of 18 ft 2. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 a. Which perfect square is less than 18 but closest to 18? Mark this integer on the number line. b. Which perfect square is greater than 18 but closest to 18? Mark this integer on the number line. c. What is the square root of the first integer you marked? Write the square root above the integer on the number line. d. What is the square root of the second integer you marked? Write the square root above that integer. e. Put an X on 18. The X should be between the two perfect squares. Is the X closer to the smaller or the larger perfect square? f. Above the X you put on 18, write a decimal number to the nearest tenth that you think is the square root of 18. g. Check your estimate by squaring it to see how close you are. Try another decimal number if your check is not close to 18. 18. Using the method you used in number 17 estimate the square root of each integer. a. 42 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 b. 98 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 Unit 1 Integers and Rational Numbers 33
ACTIVITY 1.4 Powers and Roots SUGGESTED LEARNING STRATEGIES: Summarize/ Paraphrase/Retell, Visualize, Group Presentation, Think/ Pair/Share, Look for a Pattern TECHNOLOGY You can check your work in the table with a calculator. Multiply your approximate square root by itself or use the x 2 key. 19. Choose a number from the table below and find its approximate square root. Check by multiplying your approximations to determine how close each square root approximation is. Place your answer on the class table below. Continue with other numbers until the class table below is complete. Then use values in the class table to complete this table below. Number Square root 1 1 2 3 4 2 5 6 7 8 9 3 10 11 12 13 14 15 16 4 The student council is very happy with the work the class has done on the carnival so far. The class has found the areas of the floors and the side lengths of the booths. One concept remains for the class to review before completing all the needed work. The booths do not just take up floor space; they also have volume. 34 SpringBoard Mathematics with Meaning TM Level 2
Powers and Roots ACTIVITY 1.4 SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Use Manipulatives, Interactive Word Wall, Group Presentation The diagram at the right represents a cubic foot. Its dimensions are 1 ft 1 ft 1 ft. 20. Build a solid with dimensions of 2 units 2 units 2 units. a. What is the name of the solid you just built? b. What is true about its edge lengths? 21. Aaron says that to find the volume of a cube he uses the formula l w h. Jonelle says that the formula she uses is l 3. Who uses the correct formula? Justify your answer. 22. Use a formula from Question 21. Complete this table to show the volume of each cube in exponential form and in cubic feet. Length of an Edge of a Cube (in feet) 2 4 5 6 8 9 Calculation for Finding Volume of Cube Volume of the Cube (in exponential form) 23. The exponent used in each exponential expression of volume in Question 22 is the same. This exponent is used for the volume of a cube. Using this exponent is known as cubing a number. What is the exponent used in cubing a number? Volume of the Cube (in cubic feet) Unit 1 Integers and Rational Numbers 35
ACTIVITY 1.4 Powers and Roots SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Marking the Text, Look for a Pattern, Quickwrite, Self Revision/Peer Revision 24. If you know the volume of a cube you can determine the length of the edge of the cube. The volumes of four cubes are given in this table. Find the length of the edge of each cube. Volume of the Cube (in cubic units) 1 8 27 64 Length of the Edge (in units) WRITING MATH The index of 3 in the radical sign 3 shows a cube root. 3 8 is read the cube root of 8. The operation you use to find the volume of a cube when you know the edge length is called cubing. The operation you use to find the edge length when you know the volume is called finding the cube root. The symbol used for cube root is 3. 25. Complete each table. All numbers are in standard form. Number 2 3 4 5 n Cube Number 8 27 64 125 n 3 Cube Root 26. What patterns do you see in the tables above? 27. Think about what you have discovered about the volume of a cube and finding the edge length of a cube. Write a sentence to explain what the cube root of a number means. 36 SpringBoard Mathematics with Meaning TM Level 2
Powers and Roots ACTIVITY 1.4 SUGGESTED LEARNING STRATEGIES: Self Revision/Peer Revision, Group Presentation, Think/Pair/Share, Activating Prior Knowledge, Summarize/Paraphrase/Retell, Visualize, Use Manipulatives 28. Some of the booths at the carnival will be cubical. Find the lengths of the edges of the booths listed in this chart. Volume of the Cubical Booth 216 ft 3 343 ft 3 729 ft 3 1000 ft 3 Length of each Edge of the Booth 29. Exponents other than 2 and 3 can be used. Bases can be numbers other than whole numbers. Complete this table of expressions using other exponents and bases. Number (in exponential form) 2 4 Product Using the Base as a Factor Number (in standard form) The exponent tells how many times to use the base as a factor. When the exponent is 0, the value of the expression is 1. 35 0 = 1, 575 0 = 1, n 0 = 1 1 9 9 1 ( 1 4) 3 (-4) 2 7 0 (3m) 2 30. One of the games to be played at the carnival is Expression Bingo. Jonelle s class will play the game first since they have worked so hard to plan for the carnival. To start playing, choose nine numbers from the list of Expression Bingo Numbers and write them in any order on this bingo card. Expression Bingo Numbers 5 6 7 8 10 11 19 21 23 24 25 90 91 92 93 94 95 325 350 375 400 425 Unit 1 Integers and Rational Numbers 37
ACTIVITY 1.4 Powers and Roots ORDER OF OPERATIONS 1. Grouping symbols 2. Exponents 3. Multiplication and division 4. Addition and subtraction SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Summarize/Paraphrase/Retell, Visualize, Use Manipulatives Next evaluate each expression below using order of operations rules. After you have evaluated the first problem, check to see if your answer is on your bingo card. If so, shade that box. If not, move on to the next problem. 2 0 + 3 6 6-2 + 4 1 2 6 + ( 2) 4 5 3 3 1 4(2 + 3)2 7 6 (5 + 3) 3 2 3 5 2 2 4 19 + 36 3 2 250 - (3 5) 2 The winner is the player with the most shaded boxes on his or her card. If no player completes a row, column, or diagonal, the winner is the player with the most shaded boxes on his or her card. CHECK YOUR UNDERSTANDING Write your answers on notebook paper. Show your work. 8. Evaluate each expression. your work. Be sure to include units in your a. 81 b. 3 27 answers if appropriate. 1. Write 3 3 3 3 in exponential form. 9. What is the volume of a cube with a side 2. Evaluate the power 4 3. length of 6 cm? 10. What are two whole numbers that can be substituted for n to make this statement true? 3. In science we find that some cells divide to form two cells every hour. How many cells will be formed from one cell after 7 hours? Copy and replace with =, >, or <. 4. 1 9 1 0 5. 3 4 4 3 6. If you know that 9 3 = 729, describe how to find 9 4 without having to multiply four 9 s. 7. This figure has an area of 196 in. 2 and is made up of four small squares. What is the side length of a small square? 11. MATHEMATICAL REFLECTION 9 < n < 11 As the length of the edge of a cube increases, what happens to the area of a face and the volume of the cube? Explain using diagrams. 38 SpringBoard Mathematics with Meaning TM Level 2